Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}2x+5y &= -1 \\ 2x+9y &= 3\end{align*}$
Explanation: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}-2x-5y &= 1\\ 2x+9y &= 3\end{align*}$ Add the top and bottom equations. $4y = 4$ Divide both sides by $4$ and reduce as necessary. $y = 1$ Substitute $1$ for $y$ in the top equation. $2x+5( 1) = -1$ $2x+5 = -1$ $2x = -6$ $x = -3$ The solution is $\enspace x = -3, \enspace y = 1$.